Consider the function $f$ defined on the interval $[0; +\infty[$ by $f(x) = k\mathrm{e}^{-kx}$ where $k$ is a strictly positive real number. We call $\mathcal{C}_f$ its graph in the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider point A on the curve $\mathcal{C}_f$ with x-coordinate 0 and point B on the curve $\mathcal{C}_f$ with x-coordinate 1. Point C has coordinates $(1; 0)$.
Determine an antiderivative of function $f$ on the interval $[0; +\infty[$.
Express, as a function of $k$, the area of triangle OCB and that of the region $\mathcal{D}$ bounded by the y-axis, the curve $\mathcal{C}_f$ and the segment $[OB]$.
Show that there exists a unique value of the strictly positive real number $k$ such that the area of region $\mathcal{D}$ is twice that of triangle OCB.
Consider the function $f$ defined on the interval $[0; +\infty[$ by $f(x) = k\mathrm{e}^{-kx}$ where $k$ is a strictly positive real number. We call $\mathcal{C}_f$ its graph in the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider point A on the curve $\mathcal{C}_f$ with x-coordinate 0 and point B on the curve $\mathcal{C}_f$ with x-coordinate 1. Point C has coordinates $(1; 0)$.
\begin{enumerate}
\item Determine an antiderivative of function $f$ on the interval $[0; +\infty[$.
\item Express, as a function of $k$, the area of triangle OCB and that of the region $\mathcal{D}$ bounded by the y-axis, the curve $\mathcal{C}_f$ and the segment $[OB]$.
\item Show that there exists a unique value of the strictly positive real number $k$ such that the area of region $\mathcal{D}$ is twice that of triangle OCB.
\end{enumerate}